We show interior Hölder continuity for solutions \(u\)  of (REF ) that are bounded away from 1 and this requirement is to be expected in view of the interpretation of (REF ) as a Stefan problem of the form (REF ).
Indeed, observe that this requirement for \(u\)  is equivalent to stating that the solution \(v=\phi (u)\)  is bounded.
The latter condition is needed to obtain a modulus of continuity for \(v\) , see [1]}, which is a weaker property than Hölder continuity.
Moreover, the boundedness of \(v\)  assumed in [1]} is a necessary condition; it can be deduced from assuming a modulus of continuity on \(v\) .
Returning to the original equation (REF ), we note that the condition on \(u\)  is not restrictive either; in [3]} we have shown that solutions are bounded away from 1 provided that the initial data and Dirichlet boundary conditions satisfy this assumption, which is a standard assumption in the typical problems considered in applications.
